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P83 · Packing and covering · Classic · Hard

Equilateral triangles in a square · n = 13

Pack n unit-side equilateral triangles into a square. Contact is allowed; interiors cannot overlap. Minimise the container square side.

Instancen = 13
ObjectiveMinimize container square side
Best known, unproven2.595Source-precision interval: 2.595–2.596 (values inside match)exhibited site certificate 2.59521086Compiled by Erich Friedman; Found by David W. Cantrell in July 2002. Source expression: s = 2.595+.

Formal definition

  • ContainerThe editor and certificate normalize to the unit square [0,1]². This is a coordinate convention only; scores use the literature's unit-piece scale. Boundary contact is allowed.
  • SubmissionSubmit {radius, placements:[{x,y,turn},…]}, with decimal strings of at most nine places. x,y are centres; circles require turn=0, polygons use turn=tan(θ/2). Coordinates and turn lie in ±4; 0 < radius ≤ 4.
  • ConstraintsExactly n congruent shapes lie wholly in the container with pairwise disjoint interiors. Polygon rotations are independent, not limited to quarter turns.
  • PrecisionInput decimals are exact rationals; regular-polygon vertices are algebraic, not rounded templates. Feasibility has no floating-point tolerance. A finite-decimal certificate is not a continuous optimality proof.
  • Objectivecontainer square side = 1/(2 sin(pi/3) r), smaller is better. Pages, leaderboards and literature share these units; conversion is automatic.
Open the full editor ↗
VERIFIED CONSTRUCTIONVERIFIED CONSTRUCTION

Getting a feel for it

Source and certificate

External targets are converted with their printed precision; decimal coordinate certificates are separate. Reconstruction loss never lowers the literature target, and a source construction is not automatically an optimality proof.

Source
The record-holding arrangement for Equilateral triangles in a square n = 13, 2.59521086
Current leader

2.59521086

container square side

Matches the best known
Answer sourceDavid W. Cantrell
Solution methodPublished reference construction
Challenge this record ↗Submit a proof / idea ↓Share a proof or idea in the discussion. Accepted contributions can earn proof points.
Record holder's solver noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

ANSWER FORMAT

How to write your answer

the unit square [0,1]²

Submit {radius, placements:[{x,y,turn},…]}, with decimal strings of at most nine places. x,y are centres; circles require turn=0, polygons use turn=tan(θ/2). Coordinates and turn lie in ±4; 0 < radius ≤ 4.

The current leader's answer

{
  "placements": [
    {
      "turn": "0.267949192",
      "x": "0.415130151",
      "y": "0.888766211"
    },
    {
      "turn": "-0.272346739",
      "x": "0.609616243",
      "y": "0.777539896"
    },
    {
      "turn": "0.263561294",
      "x": "0.803184104",
      "y": "0.887190735"
    },
    {
      "turn": "0.000000000",
      "x": "0.111233789",
      "y": "0.807337426"
    },
    {
      "turn": "0.577350269",
      "x": "0.222467577",
      "y": "0.614674853"
    },
    {
      "turn": "-0.556944803",
      "x": "0.882871031",
      "y": "0.584372400"
    },
    {
      "turn": "-0.267949192",
      "x": "0.475377611",
      "y": "0.444935155"
    },
    {
      "turn": "-0.000000000",
      "x": "0.111233789",
      "y": "0.422012279"
    },
    {
      "turn": "0.015440530",
      "x": "0.777638474",
      "y": "0.388367471"
    },
    {
      "turn": "-0.565441941",
      "x": "0.885325147",
      "y": "0.190634470"
    },
    {
      "turn": "-0.267949192",
      "x": "0.192662574",
      "y": "0.111233789"
    },
    {
      "turn": "0.267949192",
      "x": "0.385325147",
      "y": "0.222467577"
    },
    {
      "turn": "-0.267949192",
      "x": "0.577987721",
      "y": "0.111233789"
    }
  ],
  "radius": "0.222467576"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 13,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.267949192",
      "x": "0.415130151",
      "y": "0.888766211"
    },
    {
      "turn": "-0.272346739",
      "x": "0.609616243",
      "y": "0.777539896"
    },
    {
      "turn": "0.263561294",
      "x": "0.803184104",
      "y": "0.887190735"
    },
    {
      "turn": "0.000000000",
      "x": "0.111233789",
      "y": "0.807337426"
    },
    {
      "turn": "0.577350269",
      "x": "0.222467577",
      "y": "0.614674853"
    },
    {
      "turn": "-0.556944803",
      "x": "0.882871031",
      "y": "0.584372400"
    },
    {
      "turn": "-0.267949192",
      "x": "0.475377611",
      "y": "0.444935155"
    },
    {
      "turn": "-0.000000000",
      "x": "0.111233789",
      "y": "0.422012279"
    },
    {
      "turn": "0.015440530",
      "x": "0.777638474",
      "y": "0.388367471"
    },
    {
      "turn": "-0.565441941",
      "x": "0.885325147",
      "y": "0.190634470"
    },
    {
      "turn": "-0.267949192",
      "x": "0.192662574",
      "y": "0.111233789"
    },
    {
      "turn": "0.267949192",
      "x": "0.385325147",
      "y": "0.222467577"
    },
    {
      "turn": "-0.267949192",
      "x": "0.577987721",
      "y": "0.111233789"
    }
  ],
  "radius": "0.222467576"
}

Submit {radius, placements:[{x,y,turn},…]}, with decimal strings of at most nine places. x,y are centres; circles require turn=0, polygons use turn=tan(θ/2). Coordinates and turn lie in ±4; 0 < radius ≤ 4. · Verifier v1.0.0

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